Optimal. Leaf size=123 \[ \frac {4 c (d+e x)^{3/2} \left (a e^2+3 c d^2\right )}{3 e^5}-\frac {8 c d \sqrt {d+e x} \left (a e^2+c d^2\right )}{e^5}-\frac {2 \left (a e^2+c d^2\right )^2}{e^5 \sqrt {d+e x}}+\frac {2 c^2 (d+e x)^{7/2}}{7 e^5}-\frac {8 c^2 d (d+e x)^{5/2}}{5 e^5} \]
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Rubi [A] time = 0.05, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {697} \begin {gather*} \frac {4 c (d+e x)^{3/2} \left (a e^2+3 c d^2\right )}{3 e^5}-\frac {8 c d \sqrt {d+e x} \left (a e^2+c d^2\right )}{e^5}-\frac {2 \left (a e^2+c d^2\right )^2}{e^5 \sqrt {d+e x}}+\frac {2 c^2 (d+e x)^{7/2}}{7 e^5}-\frac {8 c^2 d (d+e x)^{5/2}}{5 e^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin {align*} \int \frac {\left (a+c x^2\right )^2}{(d+e x)^{3/2}} \, dx &=\int \left (\frac {\left (c d^2+a e^2\right )^2}{e^4 (d+e x)^{3/2}}-\frac {4 c d \left (c d^2+a e^2\right )}{e^4 \sqrt {d+e x}}+\frac {2 c \left (3 c d^2+a e^2\right ) \sqrt {d+e x}}{e^4}-\frac {4 c^2 d (d+e x)^{3/2}}{e^4}+\frac {c^2 (d+e x)^{5/2}}{e^4}\right ) \, dx\\ &=-\frac {2 \left (c d^2+a e^2\right )^2}{e^5 \sqrt {d+e x}}-\frac {8 c d \left (c d^2+a e^2\right ) \sqrt {d+e x}}{e^5}+\frac {4 c \left (3 c d^2+a e^2\right ) (d+e x)^{3/2}}{3 e^5}-\frac {8 c^2 d (d+e x)^{5/2}}{5 e^5}+\frac {2 c^2 (d+e x)^{7/2}}{7 e^5}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 97, normalized size = 0.79 \begin {gather*} -\frac {2 \left (105 a^2 e^4+70 a c e^2 \left (8 d^2+4 d e x-e^2 x^2\right )+3 c^2 \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )\right )}{105 e^5 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 123, normalized size = 1.00 \begin {gather*} \frac {2 \left (-105 a^2 e^4-210 a c d^2 e^2-420 a c d e^2 (d+e x)+70 a c e^2 (d+e x)^2-105 c^2 d^4-420 c^2 d^3 (d+e x)+210 c^2 d^2 (d+e x)^2-84 c^2 d (d+e x)^3+15 c^2 (d+e x)^4\right )}{105 e^5 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 117, normalized size = 0.95 \begin {gather*} \frac {2 \, {\left (15 \, c^{2} e^{4} x^{4} - 24 \, c^{2} d e^{3} x^{3} - 384 \, c^{2} d^{4} - 560 \, a c d^{2} e^{2} - 105 \, a^{2} e^{4} + 2 \, {\left (24 \, c^{2} d^{2} e^{2} + 35 \, a c e^{4}\right )} x^{2} - 8 \, {\left (24 \, c^{2} d^{3} e + 35 \, a c d e^{3}\right )} x\right )} \sqrt {e x + d}}{105 \, {\left (e^{6} x + d e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 137, normalized size = 1.11 \begin {gather*} \frac {2}{105} \, {\left (15 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{2} e^{30} - 84 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{2} d e^{30} + 210 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{2} d^{2} e^{30} - 420 \, \sqrt {x e + d} c^{2} d^{3} e^{30} + 70 \, {\left (x e + d\right )}^{\frac {3}{2}} a c e^{32} - 420 \, \sqrt {x e + d} a c d e^{32}\right )} e^{\left (-35\right )} - \frac {2 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} e^{\left (-5\right )}}{\sqrt {x e + d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 106, normalized size = 0.86 \begin {gather*} -\frac {2 \left (-15 c^{2} x^{4} e^{4}+24 c^{2} d \,x^{3} e^{3}-70 a c \,e^{4} x^{2}-48 c^{2} d^{2} e^{2} x^{2}+280 a c d \,e^{3} x +192 c^{2} d^{3} e x +105 a^{2} e^{4}+560 a c \,d^{2} e^{2}+384 c^{2} d^{4}\right )}{105 \sqrt {e x +d}\, e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 121, normalized size = 0.98 \begin {gather*} \frac {2 \, {\left (\frac {15 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{2} - 84 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{2} d + 70 \, {\left (3 \, c^{2} d^{2} + a c e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 420 \, {\left (c^{2} d^{3} + a c d e^{2}\right )} \sqrt {e x + d}}{e^{4}} - \frac {105 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}}{\sqrt {e x + d} e^{4}}\right )}}{105 \, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 128, normalized size = 1.04 \begin {gather*} \frac {2\,c^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5}-\frac {\left (8\,c^2\,d^3+8\,a\,c\,d\,e^2\right )\,\sqrt {d+e\,x}}{e^5}-\frac {2\,a^2\,e^4+4\,a\,c\,d^2\,e^2+2\,c^2\,d^4}{e^5\,\sqrt {d+e\,x}}+\frac {\left (12\,c^2\,d^2+4\,a\,c\,e^2\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,e^5}-\frac {8\,c^2\,d\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 17.78, size = 126, normalized size = 1.02 \begin {gather*} - \frac {8 c^{2} d \left (d + e x\right )^{\frac {5}{2}}}{5 e^{5}} + \frac {2 c^{2} \left (d + e x\right )^{\frac {7}{2}}}{7 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (4 a c e^{2} + 12 c^{2} d^{2}\right )}{3 e^{5}} + \frac {\sqrt {d + e x} \left (- 8 a c d e^{2} - 8 c^{2} d^{3}\right )}{e^{5}} - \frac {2 \left (a e^{2} + c d^{2}\right )^{2}}{e^{5} \sqrt {d + e x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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